Scalar Levin-Type Sequence Transformations

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or partial sums $s_n$ but also on an auxiliary sequence of so-called remainder estimates $\omega_n$ are of Levin-type if they are linear in the $s_n$, and nonlinear in the $\omega_n$. Known Levin-type sequence transformations are reviewed and put into a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math., abstract shortene

An approximative calculation of the fractal structure in self-similar tilings

Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the fractal dimension by using the distribution without huge computations. This method can be applied to self-similar tilings based on a stochastic process.Comment: 5 pages, 6 figure

Stability of solutions of the Sherrington-Kirkpatrick model with respect to replications of the phase space

We use real replicas within the Thouless, Anderson and Palmer construction to investigate stability of solutions with respect to uniform scalings in the phase space of the Sherrington-Kirkpatrick model. We show that the demand of homogeneity of thermodynamic potentials leads in a natural way to a thermodynamically dependent ultrametric hierarchy of order parameters. The derived hierarchical mean-field equations appear equivalent to the discrete Parisi RSB scheme. The number of hierarchical levels in the construction is fixed by the global thermodynamic homogeneity expressed as generalized de Almeida Thouless conditions. A physical interpretation of a hierarchical structure of the order parameters is gained.Comment: REVTeX4, 22 pages, second extended version to be published in Phys. Rev.

Fine-To-Coarse Global Registration of RGB-D Scans

RGB-D scanning of indoor environments is important for many applications, including real estate, interior design, and virtual reality. However, it is still challenging to register RGB-D images from a hand-held camera over a long video sequence into a globally consistent 3D model. Current methods often can lose tracking or drift and thus fail to reconstruct salient structures in large environments (e.g., parallel walls in different rooms). To address this problem, we propose a "fine-to-coarse" global registration algorithm that leverages robust registrations at finer scales to seed detection and enforcement of new correspondence and structural constraints at coarser scales. To test global registration algorithms, we provide a benchmark with 10,401 manually-clicked point correspondences in 25 scenes from the SUN3D dataset. During experiments with this benchmark, we find that our fine-to-coarse algorithm registers long RGB-D sequences better than previous methods

Fast Iterative 3D Mapping for Large-Scale Outdoor Environments with Local Minima Escape Mechanism

This paper introduces a novel iterative 3D mapping framework for large scale natural terrain and complex environments. The framework is based on an Iterative-Closest-Point (ICP) algorithm and an iterative error minimization mechanism, allowing robust 3D map registration. This was accomplished by performing pairwise scan registrations without any prior known pose estimation information and taking into account the measurement uncertainties due to the 6D coordinates (translation and rotation) deviations in the acquired scans. Since the ICP algorithm does not guarantee to escape from local minima during the mapping, new algorithms for the local minima estimation and local minima escape process were proposed. The proposed framework is validated using large scale field test data sets. The experimental results were compared with those of standard, generalized and non-linear ICP registration methods and the performance evaluation is presented, showing improved performance of the proposed 3D mapping framework